In this course students learn the basic concepts of acoustics and electronics and how they can applied to understand musical sound and make music with electronic instruments. Topics include: sound waves, musical sound, basic electronics, and applications of these basic principles in amplifiers and speaker design.
Oscillations in Space and Time
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From the course by University of Rochester
Fundamentals of Audio and Music Engineering: Part 1 Musical Sound & Electronics
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University of Rochester
287 ratings
From the lesson
Week 1 - Introduction RC & Introduction to Electrical Circuits MB
Introduction to oscillations and sound waves, simple oscillating systems, sound pressure, sound waves, the speed of sound, wavelength, frequency and pitch, sound pressure level, loudness, making sound, properties of musical sound versus “noise”. Electronics fundamentals - charge, current, voltage, power, resistance, Ohm’s law, DC circuits, finding currents and voltages in simple circuits.
Meet the Instructors
Robert ClarkProvost and Senior Vice President for Research and Professor
Mechanical Engineering
Mark BockoDistinguished Professor and Chair
Electrical and Computer Engineering
Before we really dive into the details of, you know, things like speaker design.
it's really important to understand a little bit more about wave propagation in
particular. And so I've got a, a, a diagram here, I
guess that shows this is, it turns out, this is a pinned boundary being that I
had from another problem I worked sometime ago.
But it could just as easily be a string, fixed and attached at both ends.
And what we want to do is look at the response of this string if it's plucked.
And so, what you see here so here's the boundaries sorry, here, here's the
boundaries, right here, of the string. And if we were to grab hold of the string
and pluck it. particularly if we did so at the center,
you can see this oscillatory behavior here.
That where the string is vibrating up and down basically.
And so, you get a picture that there's a wave propagation in space between the
boundaries of the of the fixed points of the string.
And that's typically what's called a mode of the string or sometimes the standing
wave on the string. the other thing that's important to think
about is the time domain response. Because if you pluck the string, at some
point the oscillations will subside, and if we were to look at the, what we'd,
what I'd label the temporal picture here you can see the time domain response of
the string. And it starts out with a high amplitude,
as you displace it of course, and then basically oscillates until until it damps
out or the, the vibration of the string ceases.
so the sensor that we placed on the string, again right here, gives us a
picture of what happens with the string at that particular point in space.
And we could move that sensor to any other point on the structure, and the
amplitude of the response, the, the peak that you see here would reflect the peak
response that you see on the wave. And so for, for this sensor, it
corresponds to the response of the string through this slice of space, if you will.
If we had put the sensor here, you would see a corresponding response that look
very similar, but with reduced amplitude, because the strings doesn't reach as
large a vibration displacement. So, anyway, just trying to tie together
the fact that there is a spatial response to wave propagation in air or on a
structure string. And then there is the temporal response
that we can measure at any point in time. And, we're going to talk about how those
two things relate to one another. So, for the spatial picture, we tend to
talk about wavelength. wavelength and spatial frequency.
wavelength represented by lambda here. Spatial frequency by new, and a wave
number k, okay? And so, the spatial frequency is related
to the wavelength as such, and shown here.
And of course, the wave number itself is simply 2 pi times 1, 2 pi times 1 over
the wavelength. or it could be represented as 2 pi times
our spatial frequency, all right. And so, it really describes the wave
number describes the number of wavelengths that occur in 2 pi units.
And so, what I've done here is plotted out, a sign wave.
and of course this is 2 pie, and so it turns out that what I've shown here is
one single wave length over the domain of 2 pie.
Now for the temporal picture, so let's talk about the time domain, remember we
talked about the spatial response that we see, the way propagation in space like
the plucked string, now we can talk about the temporal response.
And the temporal response has a period of oscillation T, all right?
And T is related to, so T is, T is our period of, of oscillation.
F is our frequency of oscillation and we have a circular frequency.
And the frequency of oscillation is related to the period 1 on T.
So, this is the time required for the one complete one complete period of the
cycle. And then, of course, omega is what's
called the circular frequency. And that's equal to 2 pi times 1 on T,
which also, omega equals 2 pi times f. And what I'd like you to to recognize, is
the analogy between the temporal picture and the spatial picture.
If you remember in the spatial picture, and let me just slide back up to the top
of the slide here. we see that k is related to 2 pi on
lambda, which equals 2 pi times nu. And if we go down to the bottom here, we
see omega equals 2 pi on T, and it also equals 2 pi f.
So, the point is, is that omega is analogous, analogous to k, our wave
number. And lambda in the spatial domain is
analogous to T, the period and the time domain.
And then of course, f, our frequency, and the time domain, is analgous to nu here,
alright, in the spatial domain. So, we're describing a wave, basically,
either in space or time. And we have different different symbols
to represent the two, but they're they're analogous to each other, okay.
And so, I can do the same thing with with in the time domain I can represent that
same [UNKNOWN] response characteristic, and this just happens to be one period of
oscillation. So, the amount of time it takes here is
one period of oscillation for this particular wave, all right, to propagate.
Now, when we move to sound wave propagation in air.
we can easily relate to wavelength to the frequency.
And that's done through the speed of sound.
And so, in air in particular. And so, before we get into those details
we need to define a few basics. So, we need to define our, you know,
we'll do it at room temperature that's what we're going to be working at,
especially around the concept of speaker design and typical acoustics in the
musical range. but speed of sound in air is roughly 340
meters per second 343 at 20 degrees c. And then the density of air, which is
important as well for some of the calculations we make, is 1.21 kilograms
per meter cubed. And you notice that I'm working in the
metric units here. Alright.
So, the relationship between wavelength and frequency of sound and air is fairly
simple. And it's expressed here, lambda, the
wavelength of the frequency. the wavelength, special wavelength, is
related to the speed of sound in air. Remember that's, that's in meters per
second. And frequency is m1 over seconds, so, you
know, if you look at the units on this, you have meters per second divided by 1
over second, which gives you units of meters, all right?
And that makes sense, because the wavelength is going to be represented in
a measure of meters. so let's talk a little bit about the
relationship between, our audible range, and then of course the the wavelength of
sound in the audible range. So, humans actually hear sound in
wavelengths that range from about 20 hertz at the low frequency to 20
kilohertz at high frequency. And you know the older you get actually
you get you, your, your response at higher frequencies decreases.
we know that but, but It's but at birth basically you have a, a range of about 20
hertz to about 20 kilohertz, and it varies from person to person depending on
auditory capabilities. So, you've just computed the wavelength
of sound at 20 hertz. what is the wave length of sound at 20
kilohertz? It's pretty simple all you have to do is
divide that by a thousand. and if you do that you get a wave length
of 1.7 centimeters. So, if our audible range is from 20 hertz
to 20 kilohertz, that's the frequency range over which we hear it corresponds
to wavelengths that range from 17 meters to 1.7 centimeters.
So, the wavelength the sound at 20 hertz is 17 meters long.
Now that's, that's, that's a large wavelength.
If you think about the, look around your surroundings at your room most of you
probably aren't in a room at this point in time that has a dimension of 17 meters
in any given dimension. So, this a really long wavelength,
particularly compared to the size of your head and the distance between your ears.
at high frequency, though at 20 kilohertz for example, wavelengths only 1.7
centimeters long, so it's very short. So, this is important because it's, it's
important in particular in relation to how we actually localize sound.
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